- #1
Ric72
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When considering the solution u(x,y) of the poisson equation
u_xx + u_yy = -1 for (x,y) in G
on a 2-dimensional domain G with Dirichlet boundary conditions
u = 0 for (x,y) on boundary of G
I am wondering the following: for what shape of the domain G do I obtain the largest area-average for the solution?
Since this answers the questions of (i) lowest pressure drop in a pipe of given cross-sectional area, (ii) which shape G of a uniformly hot-heating plate has the largest average temperature and (iii) what shape of domain has the largest average exit-time distribution for drunk sailors (random walkers) I am pretty sure someone has figured this out already.
Also, I am pretty sure the answer is a circle. But I lack a proof (or at least an authoritative answer ;-)
Thanks for your help,
Ric
u_xx + u_yy = -1 for (x,y) in G
on a 2-dimensional domain G with Dirichlet boundary conditions
u = 0 for (x,y) on boundary of G
I am wondering the following: for what shape of the domain G do I obtain the largest area-average for the solution?
Since this answers the questions of (i) lowest pressure drop in a pipe of given cross-sectional area, (ii) which shape G of a uniformly hot-heating plate has the largest average temperature and (iii) what shape of domain has the largest average exit-time distribution for drunk sailors (random walkers) I am pretty sure someone has figured this out already.
Also, I am pretty sure the answer is a circle. But I lack a proof (or at least an authoritative answer ;-)
Thanks for your help,
Ric
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